# A representation theorem of harmonic functions

I am trying to solve a problem in harmonic functions in Rudin's book(Real and Complex analysis 3rd edition)

To clarify the problem I want to ask, we need some notations:

(1) $U$ is the open unit disc, and $T$ is the unit circle, the boundary of $U$ in the complex plane

(2) $P(z,e^{it})$ is the Poisson Kernel. $$P(z,e^{it})=\frac{1-|z|^2}{|e^{it}-z|^2}$$ for $z\in U$, $e^{it}\in T$

(3)$P[f]$ is the Poisson Integral against $f\in L^1(T)$

(4)$P[d\mu]$ is the Poisson Integral against a complex measure on $T$, defined by $$P[d\mu](z)=\int_T P(z,e^{it})d\mu(e^{it})\quad (z\in U)$$

(5)$C(T)$ is the space consisting of all the continuous complex functions on $T$

(6)We associate to any function $u$ in $U$ a family of functions $u_r$ on $T$, defined by $$u_r(e^{it})=u(re^{it})\quad(0\leq r<1)$$

(7)The measure $\sigma$ is defined by $\sigma=m/2\pi$, where $m$ is ordinary Lebesgue measure on $T$

(8)$||u_r||_1$ is defined by $$||u_r||_1=\int_T |u_r|d\sigma\quad(0\leq r<1)$$

The problem is:

Suppose $u$ is harmonic in $U$, and $\{u_r:0\leq r<1\}$ is a uniformly integrable subset of $L^1(T)$. Modify the proof of Theorem 11.30 to show that $u=P[f]$ for some $f\in L^1(T)$.

Before stating Theorem 11.30, one needs theorem 11.29.

Theorem 11.29: Suppose that (a)$X$ is a separable Banach space, (b)${\Lambda_n}$ is a sequence of linear functionals on $X$, (c)$sup_n||\Lambda_n||=M<\infty$

Then there is a subsequence $\{\Lambda_{n_i}\}$ such that the limit $$\Lambda x=\lim_{i\to\infty}\Lambda_{n_i} x$$ exists for every $x\in X$. Moreover, $\Lambda$ is linear, and $||\Lambda||\leq M$

Proof (Sketch): Note that $\{\Lambda_n\}$ is pointwise bounded and equicontinuous. Since each point of $X$ is a compact set, Theorem 11.29 follows from Arzela-Ascoli Theorem. Besides, it is obvious that $||\Lambda||\leq M$ and that $\Lambda$ is linear.

Theorem 11.30: Suppose $u$ is harmonic in $U$, and $$sup_{0<r<1} ||u_r||_1=M<\infty$$ It follows that there is a unique complex Borel measure $\mu$ on $T$ so that $u=P[d\mu]$

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